Narrative:

Cruising at 410; first officer (first officer) PF (pilot flying); center gave us a crossing restriction of 270. First officer started down at 1000 fpm. We were then given a late-night 'direct ZZZ' clearance and the thrust levers came back to idle. The #1 engine soon thereafter indicated engine fail which was a bit disconcerting. The egt for #1 was significantly higher than #2. I grabbed the QRH and said I was planning on running the engine fail checklist while we evaluated the engine indications. The egt at this point was above 800. I elected to do an immediate shutdown. We verified the engine and went to cutoff. The egt was just in the red and began falling after cutoff and I continued the engine fail checklist.I conveyed our engine failure condition to ATC and we were switched to approach and checked in. Based on the normal indications on #1; we talked about a restart. The first officer expressed his concern regarding restarting an engine after it had exceeded egt redline. We ultimately agreed to restart with the understanding that an immediate shutdown may be required. Considering this; I got single-engine landing performance for the [long] runway xxr in ZZZ. The xb restart went well and the engine appeared to be in a stable idle condition. During the engine in-flight start; step #10 dictates; 'engine gen switch; affected side... On;' which I did. Considering the APU was already on the bus I realized later that it might've been wise to leave it on the APU versus loading up an already questionable engine.later during the approach; the first officer applied power and the #1 began to shudder. He brought #1 back to idle and continued the intercept and approach with the #2 - #1 remained idle. We worked through the one engine inop landing checklist. I had not officially [requested priority handling] at this point. I had neglected to do that on my initial engine-out report. The first officer reminded me of this and I got it done. Considering the first indication of a problem was during descent; there was not much extra time. I didn't feel like contact with dispatch would've been beneficial as we were fine on performance and proceeding to our destination.during descent at low power settings the engine failure and restart was transparent to the passengers; according to the fas (flight attendant) later. I did not advise the fas because of the time crunch and an expectation of a relatively normal landing. In retrospect I definitely should have given them a heads-up if for no other reason than to explain the presence of fire trucks after landing. After a flawless approach and nice landing by the first officer - we had briefed no #1 reverse thrust - I taxied off the runway; shut #1 down; and taxied to gate xx for the subsequent phone calls to maintenance; fodo (flight operations duty officer); etc. We both spoke with the fas about the situation. I have appreciated the follow-up regarding the possible causes of the failure and support from the union and company alike.

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Original NASA ASRS Text

Title: B737 Captain reported an inflight engine shutdown that resulted in a precautionary landing.

Narrative: Cruising at 410; FO (First Officer) PF (Pilot Flying); Center gave us a crossing restriction of 270. FO started down at 1000 fpm. We were then given a late-night 'Direct ZZZ' clearance and the thrust levers came back to idle. The #1 engine soon thereafter indicated ENG FAIL which was a bit disconcerting. The EGT for #1 was significantly higher than #2. I grabbed the QRH and said I was planning on running the ENG FAIL checklist while we evaluated the engine indications. The EGT at this point was above 800. I elected to do an immediate shutdown. We verified the engine and went to cutoff. The EGT was just in the red and began falling after cutoff and I continued the ENG FAIL checklist.I conveyed our engine failure condition to ATC and we were switched to Approach and checked in. Based on the normal indications on #1; we talked about a restart. The FO expressed his concern regarding restarting an engine after it had exceeded EGT redline. We ultimately agreed to restart with the understanding that an immediate shutdown may be required. Considering this; I got single-engine landing performance for the [long] Runway XXR in ZZZ. The XB restart went well and the engine appeared to be in a stable idle condition. During the Engine In-Flight Start; step #10 dictates; 'Engine GEN Switch; affected side... ON;' which I did. Considering the APU was already on the bus I realized later that it might've been wise to leave it on the APU versus loading up an already questionable engine.Later during the approach; the FO applied power and the #1 began to shudder. He brought #1 back to idle and continued the intercept and approach with the #2 - #1 remained idle. We worked through the One Engine Inop Landing Checklist. I had not officially [requested priority handling] at this point. I had neglected to do that on my initial engine-out report. The FO reminded me of this and I got it done. Considering the first indication of a problem was during descent; there was not much extra time. I didn't feel like contact with Dispatch would've been beneficial as we were fine on performance and proceeding to our destination.During descent at low power settings the engine failure and restart was transparent to the passengers; according to the FAs (Flight Attendant) later. I did not advise the FAs because of the time crunch and an expectation of a relatively normal landing. In retrospect I definitely should have given them a heads-up if for no other reason than to explain the presence of fire trucks after landing. After a flawless approach and nice landing by the FO - we had briefed no #1 reverse thrust - I taxied off the runway; shut #1 down; and taxied to Gate XX for the subsequent phone calls to Maintenance; FODO (Flight Operations Duty Officer); etc. We both spoke with the FAs about the situation. I have appreciated the follow-up regarding the possible causes of the failure and support from the union and company alike.

Data retrieved from NASA's ASRS site and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.